Dynamics and geometry from high dimensional data

Data-driven discovery of dynamical systems in the engineering, physical and biological sciences

Nathan Kutz

University of Washington

We demonstrate that we can use emerging, large-scale time-series data from modern sensors to directly construct, in an adaptive manner, governing equations (differential and partial differential equations), even nonlinear dynamics, that best model the system measured using sparsity-promoting techniques. Our sparse identification of nonlinear dynamics (SINDy) algorithm can be integrated with principles of model selection which allows for the consideration of a combinatorially large number of candidate models governing a dynamical system. The innovation circumvents a disadvantage of standard model selection which typically limits the number candidate models considered due to the intractability of computing information criteria. Using SINDy, the sub-selection of candidate models near the Pareto frontier allows for a tractable computation of AIC (Akaike information criteria) or BIC (Bayes information criteria) scores for the remaining candidate models. The information criteria hierarchically ranks the most informative models, enabling the automatic and principled selection of the model with the strongest support in relation to the time series data. Specifically, we show that AIC scores place each candidate model in the strong support, weak support or no support category, thus allowing us to select the best model for a given set of time series data. The methodology generalizes to the discovery of nonlinear PDEs as well.